Exercise 12, § 6

Let $A$ be a star algebra, $N$ the set of normal elements of $A$, $x\in N$ and $V$ a neighbourhood of 0 in $\mathbf{C}$. There exists a neighbourhood $U$ of $x$ in $N$ such that, for every $y\in U$, one has Sp$_A(y)\subset$ Sp$_A(x) + V$ and Sp$_A(x)\subset$ Sp$_A(y) + V$. (Use the cor. of 1 of I, p. 108, the prop. 3 of I, p. 23 and the prop. 10 of I, p. 76.)